Secant varieties of toric varieties

Abstract

Let $X_P$ be a smooth projective toric variety of dimension $n$ embedded in $\PP^r$ using all of the lattice points of the polytope $P$. We compute the dimension and degree of the secant variety $\Sec X_P$. We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties $X_A$ embedded using a set of lattice points $A \subset P\cap\ZZ^n$ containing the vertices of $P$ and their nearest neighbors.

v1, AMS LaTex, 5 figures, 25 pages; v2, reference added; v3, This is a major rewrite. We have strengthened our main results to include a classification of smooth lattice polytopes P such that Sec X_P does not have the expected dimension. (See Theorems 1.4 and 1.5.) There was also a considerable amount of reorganization, and some expository material was eliminated; v4, 28 pages, minor corrections, additional and updated references

References in corpus (2)

• Phylogenetic Algebraic Geometry
• Toric manifolds with degenerate dual variety and defect polytopes

Cited by in corpus (8)

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• Combinatorial secant varieties
• Sparse effective membership problems via residue currents
• Arithmetic properties of the first secant variety to a projective variety
• The secant line variety to the varieties of reducible plane curves